Applications of the updated PTQM
theory profited, first of all, from
the factorization
of metric.
Simultaneously, these applications suffered
from the loss of a direct contact
with the principle of correspondence
as offered, initially, by Eq. (REF ).
In our recent brief note [1]} we
addressed the problem.
We pointed out
that the somewhat mysterious PTQM-mediated charge-based
relegation of the Hermiticity
from the physical Hilbert space \({\cal H}_{phys}\)  to
auxiliary \({\cal H}_{math}\) 
may be given a very natural conceptual background
when one decides to treat,
on equal footing,
the observability of the Hamiltonian \(H\) 
and the observability of the charge \({\cal C}\) .
